Finite spectral triples for the fuzzy torus
FFinite spectral triples for the fuzzy torus
John W. Barrett, James GauntSchool of Mathematical SciencesUniversity of NottinghamUniversity ParkNottingham NG7 2RD, UKE-mail [email protected]@gmail.comAugust 19th, 2019
Abstract
Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integerparameters. Each of these geometries has four different Dirac opera-tors, corresponding to the four unique spin structures on a torus. Thespectrum of the Dirac operator is calculated. It is given by replacingintegers with their quantum integer analogues in the spectrum of thecorresponding commutative torus.
A Riemannian spin geometry can be expressed in algebraic terms using thealgebra of smooth functions on the manifold and the Dirac operator on thespinor bundle. This point of view allows for a significant generalisation ofgeometry by removing the restriction that the algebra is commutative, ad-justing the axioms in as natural a way as possible. The mathematical struc-ture that results is called a real spectral triple [1] and encompasses bothcommutative and non-commutative geometries.1 a r X i v : . [ m a t h . QA ] A ug mong the non-commutative geometries, it is of significant interest toexamine examples that are analogues of Riemannian spin manifolds, or,more specifically, approximations of Riemannian spin manifolds. This pa-per contributes to this by defining and studying the properties of some non-commutative analogues of the flat torus called fuzzy tori. These analoguesare all finite in the sense that the algebra is a (finite-dimensional) matrixalgebra, and the Dirac operator is also a matrix. This notion of finitenessturns out to be richer than the corresponding notion in commutative geom-etry, which would just be the study of finite sets of points.The flat torus is obtained by identifying the opposite sides of a parallel-ogram in Euclidean space. While this is simple as a geometry, it providesinteresting examples due to the free parameters in the shape of the torus andthe four possible spin structures. Both the geometry and the spin structurecan be read off from the spectrum of the Dirac operator.A fuzzy torus is a finite non-commutative analogue of a flat torus andits geometry is specified by a choice of two elements of its algebra. Themain result of this paper is the construction of a Dirac operator for thisfuzzy torus geometry, forming a real spectral triple. The general formula isanalogous to the commutative formula for a Dirac operator constructed witha particular frame field. In simple examples, where the fuzzy torus algebra isa simple matrix algebra, the spectrum of the Dirac operator is a deformationof the spectrum of a commutative Dirac operator and one can determine thecorresponding spin structure of the commutative torus from the spectrum.This coincides with the spin structure determined by the frame field.In further examples with a non-simple algebra, it is possible to constructDirac operators that correspond to all four possible spin structures on thecommutative torus. This is done by constructing the non-commutative ana-logue of a four-fold covering of the torus. The torus
The algebra of functions on the torus T is the commutativealgebra generated by U = e iθ , V = e iφ . (1)Defining X = e i ( aθ + bφ ) , Y = e i ( cθ + dφ ) , (2)for integers a, b, c, d , the principal objects of study are differential operatorsspecified using X and Y . These correspond to a Riemannian metric on thetorus. In the simplest case X = U and Y = V , this is the square metricd θ + d φ . (3)2n the general case, the metric is the pull-back of this by the covering map( θ, φ ) (cid:55)→ ( aθ + bφ, cθ + dφ ) . (4)The finite non-commutative torus is the study of unitary matrices U and V , subject to the relation U V = qV U (5)for a complex number q , which has to be a root of unity. This is simultane-ously a deformation of the torus and a truncation to finite dimensions.The fuzzy torus is the study of differential operators on this algebra thatare constructed using monomials X and Y in the preferred ‘coordinates’ U and V . These operators are the non-commutative analogues of the corre-sponding torus metrics.The simplest example of the relation is given by U = C and V = S , thewell-known clock and shift matrices, which generate the algebra M N ( C ) of all N × N matrices. More generally, one can take U and V to be monomials in C and S . A particularly important example is given by U = C and V = S .The algebra generated by U and V is a subalgebra of M N ( C ), giving a non-commutative version of the regular covering map ( θ, φ ) (cid:55)→ (2 θ, φ ). Thecharacters of the deck transformation group Z × Z distinguish the differentspin structures on the base of the covering. In the non-commutative case,the analogous construction allows one to define fuzzy tori with different spinstructures. Spectral triples
The idea of a spectral triple is that it provides a math-ematical framework for a Dirac operator. A commutative spectral tripleconsists of a commutative algebra A , a Hilbert space H on which A acts,and a Dirac operator D : H → H . For a Riemannian spin manifold M , A isthe algebra of smooth functions on M , H is the square-integrable sectionsof the spin bundle, and D is the usual Dirac operator. The spin structuredefines an antilinear map J : H → H , called the real structure.In the non-commutative generalisation [2], A is obviously a non-commu-tative algebra. Unlike the commutative case, this allows for a distinct rightaction of A on H , promoting it to a bimodule rather than just a left module.The real structure plays the role of interchanging the left and right actions. The results
The main result of this paper is the construction of finite realspectral triples that are non-commutative analogues of various metrics for thetorus, for all four spin structures. This is done by extending the constructionof the fuzzy torus. 3he Dirac operator is constructed using algebra elements X and Y in Def-inition 11. It is a very close non-commutative analogue of the commutativeone, with derivatives replaced by commutators and functions replaced by an-ticommutators. The non-trivial part of the construction, in the commutativecase, is that it is necessary to use a rotating frame in the description of theDirac operator. In this case, the spin connection coefficients are non-zero.This is necessary because there is no matrix analogue of the vector fields ∂/∂θ and ∂/∂φ that would have to be used for a non-rotating frame [6]. Forexample, a matrix analogue of the vector field ∂/∂θ for the algebra generatedby clock and shift matrices C and S would be a matrix L satisfying[ L, C ] = iC, [ L, S ] = 0 . (6)However, no such matrix exists.The non-commutative version of the Dirac operator has a term that is adirect analogue of the connection coefficients. The role of the rotating frameis somewhat mysterious from the conceptual point of view, but it is worthnoting that it is one of the factors that determines the spin structure.The eigenvalues of the Dirac operator are calculated and it is found, ratherbeautifully, that these are the q -number analogues of the eigenvalues for thecommutative case, a result already known [7] for the rather simpler Laplaceoperator [8]. In fact, the spectrum of the fuzzy Dirac operator is exactly theset of square roots of the spectrum of the corresponding fuzzy Laplacian. Itis worth noting that the correct definition of the non-commutative analoguesof the connection coefficients is essential to obtain this result. Note that theconstruction of a Dirac operator on a fuzzy torus in [7] was limited to thederivative (commutator) part only and does not appear to give a spectrumthat corresponds to the commutative case.The results are quite different to existing constructions of the Dirac op-erator on the rational non-commutative torus. The algebra of the rationalnon-commutative torus can be understood as the space of sections of a bundleof matrix algebras over the torus, as explained in [9] and [20, Section 12.2].Dirac operators on the rational non-commutative torus are constructed in[10]; see also [11] for further details. They each have the same spectrum asa Dirac operator in the commutative case and the linear growth of eigenval-ues is rather different to the periodic q -number spectra of the finite spectraltriples presented here. The different spin structures on the rational non-commutative torus are understood in terms of two-fold coverings in [12],using a different covering for each spin structure rather than the universalfour-fold covering developed here. However, the overall idea of using a non-trivial character on a covering space to change spin structure is the same.4 ummary The relevant geometry of a flat torus is summarised in Section2. The non-commutative torus is described in Section 3, restricted to thefinite case, including the analogues of translations of the coordinates and themodular transformations. There are also some heuristics for the commutativelimit that guide the ideas in later sections.The fuzzy torus is introduced in Section 4. The notion of a fuzzy spaceis slightly more sophisticated than that usually considered in the literature.Instead of considering just an algebra with distinguished generators, the fuzzytorus also has a bimodule over the algebra. Of course this includes thestandard case where the bimodule is the algebra itself, but it also allows fora significant generalisation where the bimodule is different, correspondingto non-trivial line bundles over the torus with Z twists. These non-trivialbundles are the precursors of different spin structures in later sections. Adefinition of the Laplace operator is given and its properties are studied. Inparticular, it is shown that with suitable choices of matrices X and Y , theLaplace operator is the analogue of the commutative Laplace operator on atorus with a flat metric. The geometry of the torus can be seen from studyingthe spectrum of the fuzzy Laplacian.The Dirac operator on the flat torus is studied in Section 5 and specialisedto the torus in Section 5.5. These sections summarise the formalism of theDirac operator in a very explicit way. While the usual abstract notationin differential geometry is very concise and efficient, it hides the fact thatthere are many equivalent explicit formulas. Since the symmetries of thecommutative and non-commutative cases are different, it must be that not allexplicit formulas in the commutative case have non-commutative analogues.Thus it is necessary to present the formulas in a very particular way to allowthe correct generalisation to the non-commutative case.The main results are given in Section 6. The finite real spectral tripleis defined in Definition 11. This is first examined for the square torus, withone particular spin structure, and then generalised for other geometries offlat tori, illustrated by some plots of the spectra. Finally, it is then shownhow to generalise the results to all the four spin structures on a given fuzzytorus using the non-commutative analogue of a four-fold covering map. Itis explained how this construction appears to be a non-commutative anddiscrete analogue of Marsden-Weinstein symplectic reduction. Acknowledgements
Thanks are due to the hospitality of the Mathema-tisches Forschungsinstitute Oberwolfach during July 2018, where this workwas presented, as summarised in the meeting report [13]. Part of this workalso appears in JG’s PhD thesis [14] and the support of the University of5ottingham is acknowledged.
The torus is defined as the quotient T = R / (2 π Z ) . (7)Let A : R → R be a linear map with integer coefficients, i.e., A ∈ M ( Z ).It maps lattice points to lattice points and hence determines a map (cid:101) A : T → T . If A is invertible and A − also has integer coefficients, then (cid:101) A is invertible,and the set of all such A is written GL(2 , Z ). Note that in this case, det A andits inverse are integers, so det A = ±
1. If in addition det A = 1, the trans-formation is orientation-preserving and is called a modular transformation.The set of all elements with determinant 1 defines the group SL(2 , Z ), whichwill be called the modular group (note that this name is also commonly usedfor the quotient PSL(2 , Z ) = SL(2 , Z ) / {± } ).A flat metric on R is determined by the formula g = g µν d x µ d x ν (8)using the Einstein summation convention with constant coefficients g µν andcoordinates ( x , x ). All such metrics that are positive-definite are obtainedby the pull-back of the standard Euclidean metric δ = (d x ) + (d x ) by alinear map A ∈ GL(2 , R ), g = A • δ (9)and the coefficients are given by g µν = A ρµ A σν δ ρσ . (10)Note that the pull-back is denoted A • rather than the usual A ∗ so as notto confuse with the adjoint of the matrix A . The corresponding metric onthe torus will be denoted (cid:101) g and is called a flat torus . A flat torus has zeroRiemannian curvature but, moreover, the coordinates are chosen so that themetric coefficients are constant.The following lemma describes when (cid:101) g is equivalent to a second flat met-ric, (cid:101) g (cid:48) . Lemma 1.
The flat tori ( T , (cid:101) g ) and ( T , (cid:101) g (cid:48) ) are isometric iff there exists B ∈ GL(2 , Z ) such that g (cid:48) = B • g . 6 roof. Suppose B ∈ GL(2 , Z ) is such that g (cid:48) = B • g . Since B Z ⊂ Z , itdetermines a mapping on the quotient, (cid:101) B : T → T , and (cid:101) g (cid:48) = (cid:101) B • (cid:101) g .Conversely, suppose the two flat tori are isometric by (cid:101) B : T → T . Thenthere is an isometry B : R → R , taking g to g (cid:48) , that projects to (cid:101) B . Thisisometry is an affine map, and by composing by a translation one can assumeit is linear, i.e., B B maps lattice points to lattice points, andso B ∈ M ( Z ). Similarly, B − ∈ M ( Z ), so B ∈ GL(2 , Z ).The standard metric on the torus, (cid:101) δ , is obtained by identifying the edgesof a square with side length 2 π , and so is called a square torus. If A hasinteger coefficients, A ∈ GL(2 , R ) ∩ M ( Z ), then the metric is obtained bythe pull-back of (cid:101) δ by the map (cid:101) A . These metrics will play an important rolein the fuzzy torus and will be called integral .If g = A • δ then B • g = ( AB ) • δ . Thus the isometry class of integralmetrics on the torus is determined by the equivalence class of matrices { AB | B ∈ GL(2 , Z ) } . Each equivalence class has a unique representativecalled the Hermite normal form of the integer matrix A . These are the ma-trices A = (cid:18) a c d (cid:19) (11)with a > ≤ c < d . These matrices will be useful for constructingexamples. The orientation of the torus can be registered with a 2-form that is compatiblewith the metric, making a K¨ahler structure. For the standard Euclideanmetric, define ω = d x ∧ d x . (12)and the corresponding 2-form on the torus is denoted (cid:101) ω . The standardorientation O is the orientation of T such that (cid:90) T , O (cid:101) ω = 4 π , (13)i.e., the integral is positive.For a general metric g , given by (9), the compatible 2-form on R is thepull-back ω = A • ω , and determines a 2-form (cid:101) ω on the torus. Integratingthis gives (cid:90) T , O (cid:101) ω = (det A ) 4 π = ± (cid:90) T (cid:112) det g d x d x . (14)7he sign in this equation can be taken to indicate an orientation of the torus,positive being the standard orientation O and negative the opposite one, −O .Note that for an integral metric the factor (det A ) counts the winding numberof the map (cid:101) A .Another approach to specifying the metric is to give coordinates in whichthe metric components take a standard form. It is convenient to use theexplicit notation x = θ, x = φ. (15)Since these are not functions on the torus, one takes exponentials instead.The standard metric and orientation is specified by the ordered pair of func-tions U = e iθ , V = e iφ . (16)so that (cid:101) δ = d U d U + d V d V, (cid:101) ω = − U V d U ∧ d V. (17)For a general integral metric, if A ∈ GL(2 , R ) ∩ M ( Z ) is the matrix A = (cid:18) a bc d (cid:19) (18)the pull-back functions are X = e i ( aθ + bφ ) = U a V b , Y = e i ( cθ + dφ ) = U c V d (19)and the pull-backs of (17) now determine (cid:101) g and (cid:101) ω , (cid:101) g = d X d X + d Y d Y = ( a + c )d θ + 2( ab + cd )d θ d φ + ( b + d )d φ (20) (cid:101) ω = − XY d X ∧ d Y = ( ad − bc )d θ ∧ d φ. (21)The Poisson bracket is the inverse of the 2-form ω . The 2-form is ex-pressed as a tensor, using the (not universal) convention ω = 12 ω µν d x µ ∧ d x ν = ω µν d x µ ⊗ d x ν , (22)assuming ω µν = − ω µν . Then defining the coefficients Ω µν to be the inversematrix, so that Ω µν ω νρ = δ µρ , the Poisson bracket isΩ( f, g ) = Ω µν ∂f∂x µ ∂g∂x ν . (23)For the Poisson bracket determined by ω this formula isΩ ( f, g ) = (cid:18) ∂f∂φ ∂g∂θ − ∂f∂θ ∂g∂φ (cid:19) . (24)8pplying this to exponentials givesΩ ( U m V n , U k V l ) = ( ml − nk ) U m + k V n + l (25)In general, one hasΩ( U m V n , U k V l ) = ml − nkad − bc U m + k V n + l (26)and in particular, Ω( X, Y ) =
XY. (27)
Definition 1.
A non-commutative torus is a pair of unitary operators U and V on a Hilbert space h and a complex number q satisfying U V = qV U. (28)A finite non-commutative torus is one for which h is finite-dimensional.The unitary condition implies (by taking the adjoint of both sides) that | q | = 1. Heuristically, one can view (28) as determining a deformation of thecommutative algebra of functions on a torus.Two such geometries are regarded as equivalent if there is a unitary iso-morphism P : h → h (cid:48) transforming one to the other, P U P − = U (cid:48) , P V P − = V (cid:48) . (29)A geometry is irreducible if there is no invariant proper subspace of h .This paper considers exclusively the case of finite non-commutative tori,i.e., U and V are matrices. The basic properties were derived by Weyl [15,IV D], and Mackey [16] in more generality.If ψ is an eigenvector of U with eigenvalue λ , then V ψ has eigenvalue qλ .Since there are only a finite number of eigenvalues, it must be that q N = 1 (30)for some positive integer N . The order of q is the smallest possible value of N and then q is said to be a primitive N -th root of unity.The Hilbert space decomposes into a direct sum of irreducibles. Thisis because the orthogonal complement of an invariant subspace is also aninvariant subspace. Therefore to understand the general case it suffices tolook at the irreducibles. 9o now assume h is irreducible. The operators U N and V N commutewith each other, and with U and V , so in the λ eigenspace of U there aresimultaneous eigenvectors of both U N and V N . Let ψ be one of these. Thenthe subspace with basis { ψ, V ψ, V ψ, . . . , V N − ψ } (31)is an invariant subspace for U and V . So it follows that in an irreducible theeigenspaces of U are all one-dimensional and U N = α , V N = β , α, β ∈ C . (32)Unitarity implies | α | = | β | = 1. The numbers q, α, β characterise the irre-ducible geometry. Example 1.
The standard example is given by the clock and shift matricesof size N determined by a complex number q of order N , so that q N = 1.These are the N × N matrices C = . . . q . . .
00 0 . . . 00 0 0 q N − , S = . . .
11 0 . . .
00 . . . 0 00 0 1 0 (33)These satisfy the relations CS = q SC, C N = 1 , S N = 1 (34)So putting U = C , V = S , q = q , one finds that α = β = 1.The general irreducible case is obtained from the clock and shift matricesby a rescaling. Lemma 2 (Weyl) . Let U , V be an irreducible finite non-commutative torus.Define the clock and shift matrices C and S according to Example 1, with q = q . Then there is a basis of h such that U = α /N C , V = β /N S . Proof.
Define C = α − /N U , S = β − /N V , for an arbitrary choice of the N -throots. Using the following basis { ψ, Sψ, S ψ, . . . , S N − ψ } , (35)the matrices are those given in Example 1.10he linear maps of h to itself, End( h ), form a ∗ -algebra over C . The ∗ operation is the the adjoint of a linear map. Definition 2.
Let (
U, V, h ) be a finite non-commutative torus. The torusalgebra (cid:104) U, V (cid:105) is the ∗ -subalgebra of End( h ) generated by U and V .The torus algebra is semisimple and hence isomorphic to a direct sum ofmatrix algebras. Let N be the order of q . In the irreducible case, Lemma 2implies that (cid:104) U, V (cid:105) ∼ = M N ( C ). One can also see directly that the matrices { C n S m | m, n = 0 . . . N − } are linearly independent and so form a basis of M N ( C ). In the general case, (cid:104) U, V (cid:105) is isomorphic to the direct sum of a finitenumber of copies of M N ( C ).In the following, the matrices C and S are always the clock and shiftmatrices defined in Example 1, whereas U and V are operators defined in avariety of different ways. There are interesting examples obtained by takingmonomials in C and S . Example 2.
Let C , S be the clock and shift matrices of size N with param-eter q . Define U = C , V = S . Then U V = qV U with q = q . If N is odd,then q is again order N and the geometry is irreducible.However if N is even, then the order of q is N/
2. The geometry deter-mined by U and V reduces into the direct sum of two irreducible geome-tries according to the eigenvalues β = ± S N/ . The algebra (cid:104) U, V (cid:105) is M N/ ( C ) ⊕ M N/ ( C ).It is worth comparing the finite case with an infinite non-commutativetorus with the same value of q . This is called a rational non-commutativetorus. It is shown in [20, Prop 12.2] that the universal torus algebra isisomorphic to a bundle of matrix algebras over a commutative torus. Thiscommutative torus is generated by U N and V N and each fibre of the bundleis isomorphic to M N ( C ). Any other torus algebra (finite or infinite) is aquotient of this universal torus algebra. In the finite case, the commutativetorus is replaced by a finite subset of points determined by the eigenvaluesof U N and V N , so that the algebra is a direct sum of matrix algebras asdescribed above. For any torus algebra (cid:104)
U, V (cid:105) and a choice of square root q / one can definea convenient normalisation of the monomials of the generators that resolvesthe ambiguity between V n U m and U m V n = q mn V n U m .11 efinition 3. The normalised monomials E ( m,n ) are defined for integers m, n by E ( m,n ) = q − mn/ U m V n = q mn/ V n U m . (36)The multiplication of the monomials is E ( m,n ) E ( k,l ) = q ( ml − nk ) / E ( m + k,n + l ) , (37)so in particular, ( E ( m,n ) ) k = E ( km,kn ) (38)for any integer k .The anticommutator of two matrices is { A, B } = AB + BA . For thenormalised monomials this is { E ( m,n ) , E ( k,l ) } = ( q ( ml − nk ) / + q − ( ml − nk ) / ) E ( m + k,n + l ) . (39)which gives a commutative but non-associative product.The commutators are given by the following expression.1 q / − q − / [ E ( m,n ) , E ( k,l ) ] = q ( ml − nk ) / − q − ( ml − nk ) / q / − q − / E ( m + k,n + l ) . = [ ml − nk ] q E ( m + k,n + l ) , (40)using the quantum integer [ n ] q = q n/ − q − n/ q / − q − / (41)defined for any integer n . Suppose q = e πiK/N with K coprime to N , then q / = ± e πiK/N and [ n ] q = ( ± n − sin( πnK/N )sin( πK/N ) , (42)a form that is useful for computations.The normalised monomials also generate a finite non-commutative torus.Let a, b, c, d be integers, and define X = E ( a,b ) , Y = E ( c,d ) . (43)These obey the relation XY = q ad − bc Y X (44)Moreover, the normalised monomials in X and Y are just examples of nor-malised monomials E of U and V in the obvious way, i.e., putting Q = q ad − bc with square root Q / = q ( ad − bc ) / , then Q − kl/ X k Y l = E ( ka + lc,kb + ld ) . (45)12 lock and shift algebra The definition can be applied to the clock andshift operators C and S with a choice of q / . If the order N of q is even,either square root can be chosen and q N/ = − N odd, it is convenient to choose the root such that q N/ = +1.The normalised monomials of clock and shift are denoted e ( m,n ) to preventany confusion with more general cases. The specific feature of the clock andshift generators is that C N = S N = 1. This is preserved by the normalisedmonomials in the following sense. Lemma 3.
Let k be an integer so that kn = 0 (mod N ) and km = 0 (mod N ). Then ( e ( m,n ) ) k = 1 (46) Proof.
Using the periodicity, ( e ( m,n ) ) k = e ( km,kn ) = e (0 , = 1. Note that themiddle equality relies on the choice q N/ = 1 in the case of N odd.The monomials are periodic in n and m ; for N odd e ( m + N,n ) = e ( m,n ) , e ( m,n + N ) = e ( m,n ) , (47)while for N even e ( m + N,n ) = ( − n e ( m,n ) , e ( m,n + N ) = ( − m e ( m,n ) . (48) Two separate types of transformations of a torus algebra are defined here.Their geometric interpretation is discussed in Section 3.3.
Translations
Let U , V generate a torus algebra with q of order N . Definition 4.
For each pair of integers ( j, n ), the unitary operator P ( j,n ) = V − j U n determines the inner automorphism a (cid:55)→ P ( j,n ) a P − j,n ) (49)for all a ∈ (cid:104) U, V (cid:105) .In particular, P ( j,n ) U P − j,n ) = q j U and P ( j,n ) V P − j,n ) = q n V . These trans-formations, for all j and n , define an action of the abelian group Z N × Z N on (cid:104) U, V (cid:105) . (Here Z N ≡ Z /N Z .) It is worth noting that the P ( j,n ) considered asoperators in h do not commute. They determine a representation of a centralextension of Z N × Z N that is a type of Heisenberg group.13 odular transformations These are defined for the specific case of thealgebra M N ( C ) generated by C and S . Definition 5.
Let A = (cid:18) a bc d (cid:19) ∈ SL(2 , Z ). The matrix determines anautomorphism of M N ( C ) by T A : C (cid:55)→ e ( a,b ) , S (cid:55)→ e ( c,d ) (50)This is called a modular transformation.A calculation shows that composing these automorphisms gives T A (cid:48) ◦ T A = T AA (cid:48) . (51)These transformations can be implemented by a unitary operator acting in h according to (29). This is because all representations of the clock and shiftalgebra are equivalent, according to Lemma 2. The next two examples giveexplicit formulas for two matrices that generate SL(2 , Z ). Example 3.
Let A = (cid:18) (cid:19) . This is the transformation P A CP − A = q − / CS, P A SP − A = S. (52)An explicit formula for P A is P A = 1 √ N N − (cid:88) n =0 q n / S n , (53)and the matrix elements of P A = [( P A ) jk ], with indices j, k from 0 to N − P A ) jk = 1 √ N q ( j − k ) / . (54) Example 4.
Let B = (cid:18) −
11 0 (cid:19) . This is the transformation P B CP − B = S − , P B SP − B = C. (55)In this case, the matrix elements of P B are( P B ) jk = 1 √ N q jk , (56)which is a Vandermonde matrix. 14 .3 Correspondence This section gives some comments on the correspondence between the com-mutative torus and the finite non-commutative torus. The purpose is to guidegeometrical thinking about the non-commutative torus; the ideas about acommutative limit are mostly heuristic at this stage.The first point is that the monomials E ( m,n ) are non-commutative ana-logues of the functions e i ( mθ + nφ ) . In the commutative case ( m, n ) are arbitraryintegers and so the momentum space formed by the exponentials is the group Z × Z . However, for the simplest case of the non-commutative torus in which U and V are clock and shift matrices, the momentum space has period N according to (47) and (48), and so is a discrete torus. Considering the e ( m,n ) projectively, i.e., ignoring the phase factor, the multiplication law (37) formsa group isomorphic to Z N × Z N .Position space is dual to momentum space; the group dual to Z N × Z N isagain Z N × Z N . This suggests the finite non-commutative torus is analogousto a toroidal lattice. The translations on this lattice are the operators (49),and the eigenvectors for these operators (commutatively the plane waves) arethe monomials e ( m,n ) .The action of the modular group is very similar in the commutative andnon-commutative cases, the equation (50) being the analogue of (19). Thedifference is that for the clock and shift algebra the modular transformationsact on a toroidal lattice.The quantum integers have the following limit as q →
1. Suppose q / → σ = ±
1. Then for a fixed n [ n ] q → σ n − n. (57)Thus the q / → ml − nk ) E ( m + k,n + l ) (58)which is the analogue of the Poisson bracket Ω , according to (25).This suggests that one should think of an operator v A : B (cid:55)→ Q / − Q − / [ A, B ] (59)as an analogue of the vector field Ω( A, · ) associated to the Hamiltonian A .It satisfies a Leibnitz identity, v A ( BC ) = Bv A C + ( v A B ) C. (60)15n the case considered in (40), Q = q and so the relevant Poisson structureis Ω . In general, Q = q r and (40) generalises to1 Q / − Q − / [ E ( m,n ) , E ( k,l ) ] = [( ml − nk )] q [ r ] q E ( m + k,n + l ) (61)It is useful in places to extend Definition (41) of a quantum integer toallow a rational number a/b as argument in place of n . This can be doneas long as the definition of q / b is given. If q / b = ± e πiK/bN then the sineformula (42) again holds.In (61) one can write[( ml − nk )] q [ r ] q = (cid:20) ml − nkr (cid:21) Q , (62)where the required root of Q is defined by Q / r = q / . The fuzzy torus is a finite non-commutative torus with a Hilbert space inwhich U and V act on both the left and the right. This is achieved byadding a real structure to the non-commutative torus. Definition 6.
A real structure for a finite non-commutative torus (
U, V, h )is an antiunitary map J : h → h that is an involution, J = 1, and obeys[ J U J − , U ] = 0 , [ J U J − , V ] = 0 , [ J V J − , U ] = 0 , [ J V J − , V ] = 0 . (63)The real structure in Definition 6 determines a right action of U and V ,written for ψ ∈ h as ψU = J U ∗ J − ψ, ψV = J V ∗ J − ψ. (64)This commutes with the left action, making h a bimodule over the algebra (cid:104) U, V (cid:105) . It also implies that commutators and anticommutators with elementsof h make sense, e.g.,[ U, ψ ] =
U ψ − ψU, { U, ψ } = U ψ + ψU. (65)The right action is also unitary, i.e., ψU ∗ = ψU − , ψV ∗ = ψV − . (66)16 efinition 7. A fuzzy torus is a finite non-commutative torus with a realstructure.The Hilbert space of the fuzzy torus is the non-commutative analogue ofthe space of sections of a bundle over the torus. In the simplest case of atrivial line bundle the sections are just the complex functions and the realstructure is complex conjugation. The fuzzy analogue of this is the followingset of examples.
Example 5.
Let (
U, V, h (cid:48) ) be a finite non-commutative torus. The Hilbertspace of the fuzzy torus is h = End( h (cid:48) ), with inner product ( ψ, φ ) = Tr( ψ ∗ φ ).Then U, V ∈ End( h (cid:48) ) act by left and right multiplication, and the real struc-ture is the Hermitian conjugate, J = ∗ . These examples include the caseswhere h (cid:48) = C N , h = M N ( C ), and the multiplications are all matrix multipli-cation. The Laplace operator illustrates many of the features of the fuzzy torus in asimpler setting than the Dirac operator. One can define an analogue of theLaplace operator for the fuzzy torus and check that its eigenvalues convergeto the eigenvalues of the corresponding differential operator on the flat torus.The modular transformations are also defined for the fuzzy torus and respectthe limit.The Laplace operator described here is the analogue of the differentialLaplace operator defined on functions, or slightly more generally, sections ofa vector bundle associated to a principal bundle with a discrete group. This iscalled the scalar Laplace operator. Recall that for the commutative Laplaceoperator to be defined, a metric must be specified on the torus. The metric isindependent of the algebra of functions on the torus and thus is an additionalpiece of data that must be supplied. This additional data can be given astwo new algebra elements
X, Y ∈ C ( T ) with the metric determined by (20).Analogously, the scalar Laplace operator on a fuzzy torus is determined bytwo algebra elements X and Y satisfying XY = QY X with Q ∈ C . Let h be the bimodule of a fuzzy torus determined by U and V . Definition 8.
The scalar Laplace operator on a fuzzy torus is the operator h → h given by∆ X,Y = − Q / − Q − / ) ( [ X, [ X ∗ , · ]] + [ Y, [ Y ∗ , · ]] ) , (67)with X, Y ∈ (cid:104)
U, V (cid:105) . 17his operator is self-adjoint and so has real eigenvalues. The constructionis respected by transformations that act on h . For example, let P ∈ (cid:104) U, V (cid:105) such that
P XP − = X (cid:48) , P Y P − = Y (cid:48) . Then P acts in h by conjugation,and ( P · P − )∆ X,Y ( P − · P ) = ∆ X (cid:48) ,Y (cid:48) . Example 6.
In the simple case where h = M N ( C ), J = ∗ , X = U = C , Y = V = S and Q = q , the eigenvalues can be computed using (40),∆ C,S e ( k,l ) = ([ k ] q + [ l ] q ) e ( k,l ) . (68)In the limit q → k and l fixed,[ k ] q + [ l ] q → k + l . (69)These are the eigenvalues for the Laplacian∆ δ = − (cid:18) ∂ ∂φ + ∂ ∂θ (cid:19) (70)acting on functions on the square flat torus with metric δ , the eigenvalueequation being ∆ δ e i ( kθ + lφ ) = ( k + l ) e i ( kθ + lφ ) . (71)Looking at the spectrum as a whole, the picture is a little more compli-cated. Let q = e πiK/N , with 0 < K < N/ N , so that q hasorder N . The sine curves (42) would each have K zeroes if the argumentwere continuous; for the integer values therefore the eigenvalues are close tothe value 0 for K times as k and l cycle through their N values. However,the local minima are not all the same.For example, if N is odd and K = 2 then q / = e iπ/N and[0] q = 0 but (cid:20) N ± (cid:21) q = (cid:20) (cid:21) q . (72)Shifting the indices k and l by defining integers k (cid:48) = k − N/ − / l (cid:48) = l − N/ − / k ] q = (cid:20) k (cid:48) + 12 (cid:21) q , [ l ] q = (cid:20) l (cid:48) + 12 (cid:21) q . (73)Now taking a limit q / → k (cid:48) , l ), ( k, l (cid:48) ) or ( k (cid:48) , l (cid:48) ) leads tothe three other limits (cid:18) k (cid:48) + 12 (cid:19) + l , k + (cid:18) l (cid:48) + 12 (cid:19) , or (cid:18) k (cid:48) + 12 (cid:19) + (cid:18) l (cid:48) + 12 (cid:19) . (74)18hese spectra are characteristic of the Laplacian on line bundles over thetorus, as will be explained in Section 4.2. Note that this analysis does notwork if K is odd and fixed in the limit; in those cases q / → − / q does not converge at all.The limits in the previous example can be understood using the heuristicsfor the commutative limit from Section 3.3. For the eigenvectors C (cid:32) e iθ , S (cid:32) e iφ , and for the scaled commutator v X (cid:32) Ω ( X, · ). Then, assuming q / →
1, 1 q / − q − / [ C, · ] (cid:32) − ie iθ ∂∂φ , q / − q − / [ C ∗ , · ] (cid:32) ie − iθ ∂∂φ q / − q − / [ S, · ] (cid:32) ie iφ ∂∂θ , q / − q − / [ S ∗ , · ] (cid:32) − ie − iφ ∂∂θ . (75)With these replacements, equation (68) becomes (71). Example 7.
More generally, with the same Hilbert space and algebra asExample 6, but defining the geometry with X = e ( a,b ) , Y = e ( c,d ) , so that Q = q ad − bc , one can check that this geometry is a fuzzy version of (19).The eigenvectors and eigenvalues of ∆ X,Y are∆
X,Y e ( k,l ) = [ al − bk ] q + [ dk − cl ] q [ ad − bc ] q e ( k,l ) (76)for integers ( k, l ) ∈ Z × Z , the eigenspaces being periodic in k and l withperiod N .Using the same technique as for the square torus example, the limitingeigenvalue equation is∆ g e i ( kθ + lφ ) = ( al − bk ) + ( dk − cl ) ( ad − bc ) e i ( kθ + lφ ) . (77)with the Laplacian∆ g = − ad − bc ) (cid:18) ( b + d ) ∂ ∂θ − ab + cd ) ∂ ∂θ∂φ + ( a + c ) ∂ ∂φ (cid:19) . (78)Writing ∆ g = − g ab ∂ a ∂ b , the coefficients g ab are exactly the inverse of theintegral metric (20).In some further examples the algebra (cid:104) U, V (cid:105) may be a proper subalgebraof M N ( C ). In these cases it is possible to decompose h into subspaces thatare still bimodules over the algebra. 19 xample 8. Suppose h = M N ( C ), J = ∗ , U = X = C , and V = Y = S .The eigenvalue of eigenvector C k S l is λ k,l = [ k ] q + [2 l ] q [2] q = (cid:20) k (cid:21) q + [ l ] q (79)with Q = q = q . • For N odd, { [2 l ] q | l = 1 , . . . , N } = { [ l ] q | l = 1 , . . . , N } . Thusthe spectrum is, counterintuitively, the same as the unit square torus,except for the overall normalisation factor of [2] q . • For N even, q has order N/ l + N/ q = [ l ] q , which meansthat the l term in the eigenvalue formula ranges over the set { [ l ] q | l =1 , . . . , N/ } , with multiplicity two.The torus algebra is (cid:104) U, V (cid:105) ∼ = M N/ ( C ) ⊕ M N/ ( C ), a proper subalgebraof M N ( C ). The Hilbert space splits as a bimodule as h = h ⊕ h with h spanned by the monomials with even powers of C and h the oddpowers. Then the eigenvalues on h are λ m,l = [ m ] q + [ l ] q (80)for m = 1 , . . . , N/
2. The eigenvalues on h are λ m − ,l = (cid:20) m − (cid:21) q + [ l ] q . (81) Example 9.
Consider the fuzzy torus defined by h = M N ( C ) with N evenand J = ∗ , X = U = C , Y = V = S . Then Q = q = q , which has order N/ N/ N/ N/ C k S l is λ k,l = (cid:20) k (cid:21) q + (cid:20) l (cid:21) q . (82)The Hilbert space splits into four subspaces with even or odd powers of C and S , h = h ⊕ h ⊕ h ⊕ h , (83)with h hj spanned by monomials C k S l satisfying ( k, l ) = ( h, j ) mod 2. Notethat these subspaces can also be characterized as the eigenspaces of the ad-20oint action of C N/ and S N/ . The eigenvalues on these subspaces are h : λ m, n = [ m ] q + [ n ] q h : λ m − , n = (cid:20) m − (cid:21) q + [ n ] q h : λ m, n − = [ m ] q + (cid:20) n − (cid:21) q h : λ m − , n − = (cid:20) m − (cid:21) q + (cid:20) n − (cid:21) q . (84)for integer m, n , the eigenspaces recurring with period N/
2. The formulasfor the eigenvalues each have period N/
4, however.The torus algebra (cid:104) C , S (cid:105) has dimension N /
4. It can be characterisedas the subalgebra of M N ( C ) that commutes with C N/ and S N/ . • In the case where N/ C N/ and S N/ anti-commute and form aClifford algebra. Since q has order N/ (cid:104) C , S (cid:105) ∼ = M N/ ( C ). The left(or right) actions of C N/ and S N/ on h permute the four subspaces in(84) and also commute with the Laplacian. This shows that the fourspectra are in fact the same, the apparently different formulas beingan instance of the phenomenon noted already in Example 6. Eachbimodule h hj is isomorphic to an instance of Example 6, with data N (cid:48) = N/ C (cid:48) = C , S (cid:48) = S and q (cid:48) = q . • In the case where N/ C N/ and S N/ are central elements of A = (cid:104) C , S (cid:105) . The eigenvalues ± π hj = (1 + ( − h C N/ )(1 + ( − j S N/ ). Define the subalgebras A hj = π hj A . Then there is a splitting A = A ⊕ A ⊕ A ⊕ A . (85)There is a canonical isomorphism of A to the clock and shift algebra M N/ ( C ) = (cid:104) C (cid:48) , S (cid:48) (cid:105) given by π C k S l (cid:55)→ C (cid:48) k S (cid:48) l . The other summandsare also isomorphic to M N/ ( C ) but not in a canonical way. For exam-ple, the map a (cid:55)→ S n aS − n is an isomorphism A → A for any oddinteger n .The bimodule h is isomorphic to the algebra acting on itself, but thebimodules h hj are are not isomorphic to each other.A scatter plot for the eigenvalues (82) shows the behaviour of the mul-tiplicities for two different values of N . In the following plots it is assumed21 λ M u l t i p li c i t y Figure 1: A plot of multiplicity against eigenvalue in (82) with N = 28. λ M u l t i p li c i t y Figure 2: A plot of multiplicity against eigenvalue in (82) with N = 30.22hat q = e πi/N . The two cases considered in Figures 1 and 2 are for N/ N/ N/ k, l ∈ Z and j ∈ Z such that k ± l = ± (cid:48) ( N/ jN ). For all such k and l , the eigenvalues (82) are equal. Explicitly, λ k,l = sin (2 πk/N ) + sin (2 πl/N )sin (4 π/N )= sin (2 πk/N ) + cos (2 πk/N )sin (4 π/N ) = 1sin (4 π/N ) . (86)For this example, the feature that N/ N/ k, l that satisfy (86), but insteadthere are a large number that are nearby.The feature of a large multiplicity in one eigenvalue shown in Example 9occurs for many other examples. The lines where the eigenvalues are equalcan be seen in the contour plots shown in Section 6. Examples 8 and 9 consider splitting the Hilbert space spanned by clock andshift operators into subspaces with odd and even powers of these operators.Here, the geometry of the corresponding algebras and bimodules in the com-mutative case is explored and generalised.The functions U = e iθ and V = e iφ can be understood as pull-backs of C = e iθ and S = e iφ by the covering map ( θ, φ ) (cid:55)→ (2 θ, φ ). The Hilbert spaceof complex-valued functions on T with metric g is denoted h = L ( T , g ).This Hilbert space also splits into subspaces h hj , with h, j ∈ { , } , spannedby odd or even powers of e iθ and e iφ . A function ψ ∈ h hj is the pull-back ofa section Ψ of a complex line bundle over T , i.e., ψ = Ψ(2 θ, φ ) withΨ( θ + 2 π, φ ) = ( − h Ψ( θ, φ ) , Ψ( θ, φ + 2 π ) = ( − j Ψ( θ, φ ) . (87)The line bundle is specified by the periodic or anti-periodic boundary condi-tions determined by h and j . General case
Considering a more general case puts the about considera-tions in context and provides some results that are useful later.A map of manifolds that is a local isomorphism (i.e., an immersion ofmanifolds of the same dimension) and onto is called a covering map. Let M be a connected manifold and τ : (cid:99) M → M (88)23 regular covering map, with (cid:99) M also connected. Here, ‘regular’ means that τ • (cid:0) π ( (cid:99) M ) (cid:1) is a normal subgroup of π ( M ). Then, putting G = π ( M ) /τ • (cid:0) π ( (cid:99) M ) (cid:1) , (89)the covering is a principal G -bundle, with a right action of G on (cid:99) M . If χ is a unitary character for G (a group homomorphism G →
U(1)) thenthe associated bundle construction determines a Hermitian line bundle L χ over M . In this construction, a point of the bundle is an equivalence class[( z, ζ )] ⊂ (cid:99) M × C under the relations ( z, ζ ) ∼ ( zg, χ ( g − ) ζ ) for all g ∈ G . Thesections of L χ correspond to functions ψ on (cid:99) M such that ψ ( zg ) = χ ( g − ) ψ ( z ) for all g ∈ G . (90)Since G is a discrete group, the principal bundle (cid:99) M has a unique connec-tion on it and therefore the line bundle L χ has a uniquely-defined covariantderivative.Now suppose that G is a finite group and that M has a volume form υ .Then (cid:99) M has the volume form (cid:98) υ = |G| τ • υ . Define h = L ( (cid:99) M , (cid:98) υ ), the Hilbertspace of complex functions on the covering space. The group G has a unitaryleft action in this Hilbert space by ψ (cid:55)→ gψ with( gψ )( z ) = ψ ( zg ) . (91)Let h χ be the subspace in which G acts by the character χ , i.e.,( gψ )( z ) = χ ( g ) ψ ( z ) . (92)Comparing with (90) shows that h χ is isomorphic to L ( M , L χ , υ ), the sec-tions of the line bundle determined by the conjugate character χ ( g ) = χ ( g − ).If G is an abelian group then all the irreducible representations of G aredetermined by unitary characters and so h = (cid:77) χ ∈G ∗ h χ , (93)where G ∗ is the dual of G . This section presents the general formalism for Dirac operators on manifoldsin a way that is amenable for the construction of non-commutative analogues.24 .1 Spin structures
The definition of a spin structure is independent of the choice of metric,so this is a good place to start the discussion of spin geometry. Let M be an oriented manifold of dimension m . A frame at a point x is a linearisomorphism e x : R m → T M x preserving the orientation; the collection of allframes for all x is a principal bundle π : E → M called the (oriented) framebundle. It has structure group GL + ( m ); an element l of this group acting onframes by e x (cid:55)→ e x l. (94)The idea behind the definition of spin structures is to describe the possibleliftings of E to principal bundles with a structure group that covers GL + ( m ).This will be described more fully in Section 5.2.The inclusion SO( m ) ⊂ GL + ( m ) is a homotopy equivalence and so π (GL + ( m )) = π ( SO ( m )). For m ≥ Z and so the universalcovering group of GL + ( m ) is a 2-1 covering. For m < m ≥
3. A spin structure [19] is a cohomology class s ∈ H ( E ; Z ) having the property that the restriction to a fibre is non-trivial,i.e., if c is the non-trivial 1-cycle, then s ( c ) = 1. If spin structures exist, then M is called a spin manifold. The group H ( M ; Z ) acts freely on the set ofspin structures by s (cid:55)→ s + π • t (95)for t ∈ H ( M ; Z ), and so the number of spin structures is the number ofelements in this group.A standard result [18] is that for any connected space X , H ( X ; Z ) ∼ = Hom( H ( X ; Z ) , Z ) ∼ = Hom( π ( X ) , Z ) . (96)Therefore, a spin structure on E is equivalent to a homomorphism π ( E ) → Z . This formalism can be extended by adding extra trivial dimensions to thetangent space. The tangent space is replaced by T M x ⊕ R k , for some fixed k , and a frame is now an isomorphism e x : R m + k → T M x ⊕ R k (97)preserving an orientation on this augmented bundle. The principal bundle E is now the set of all frames of this sort. Examples are where M is immersedin an oriented manifold N of dimension m + k with a given trivialisation ofthe normal bundle. Since the inclusion GL + ( m ) ⊂ GL + ( m + k ) is a homotopyequivalence for m ≥
3, this does not change the set of spin structures.25he simplest example is to take N = M × R k . This example allows theuniform treatment of manifolds of dimensions 0 , τ : M → M (cid:48) be a covering map. It extends to a map of the framebundles and so if s (cid:48) is a spin structure on M (cid:48) , its pull-back s = τ • s (cid:48) is a spinstructure on M . Example 10.
The torus T has the frame field e x ( u, v ) = u ∂∂θ + v ∂∂φ . Sincethis is constant in x = ( θ, φ ) it is called a parallel frame field. The frame fieldis a section of the frame bundle and t = e • ( s ) ∈ H ( T ; Z ) parameterisesthe spin structures. If c and c are the 1-cycles along the two axes, then σ = (( t ( c ) , t ( c )) ∈ Z × Z is an explicit parameterisation of the four spinstructures on the torus. The spin structure s labelled by σ = (0 ,
0) is calledthe Lie spin structure and one can write s − s = π • t. (98)Let A ∈ GL + (2 , R ) ∩ M ( Z ). This determines a covering map (cid:101) A : T → T and hence a pull-back spin structure (cid:101) A • s . The key fact about the torus isthat the Lie spin structure is invariant under this pull-back. This followsfrom the fact that parallel frame fields with the same orientation are relatedby a GL + (2 , R ) transformation and so can be deformed to each other. Thus (cid:101) A • s − s = (cid:101) A • π • t = π • (cid:101) A • t (99)and so spin structure s (cid:48) = (cid:101) A • s is parameterised by t (cid:48) = (cid:101) A • t .Denote by [ n ] the integer n modulo 2. Explicitly, σ (cid:48) = (( t (cid:48) ( c ) , t (cid:48) ( c )) = (( t ( A • c ) , t ( A • c )) = (( t ( ac + cc ) , t ( bc + dc ))= ([ a ] t ( c ) + [ c ] t ( c ) , [ b ] t ( c ) + [ d ] t ( c )) = [ A T ] σ (100)with [ A T ] the transpose of the matrix A with entries modulo 2.The three spin structures (0 , ,
0) and (1 ,
1) are permuted by isomor-phisms and all of them bound a spin three-manifold.
On a Riemannian manifold the frames of the last section can be specialised toorthonormal frames, and the groups reduced to orthogonal groups and theirspin covering groups. This has the advantage that it allows the construction26f vector bundles of spinors but at the cost of introducing a dependence onthe choice of Riemannian metric.The group SO( m ) has a 2-1 covering by the homomorphism Φ : Spin( m ) → SO( m ). For m ≥
3, Spin( m ) is the universal connected cover. The groupSpin(2) is defined to be U (1) ( ∼ = SO(2)) with Φ the 2-1 cover, while Spin(1)and Spin(0) are both Z . These definitions are compatible with the stabil-isation described in the last section. While SO(2) also has r -fold covers for r >
2, the corresponding r -spin structures are not considered here.Now suppose that M is a Riemannian manifold with an orientation.There is a sub-bundle O of E consisting of the oriented orthonormal frames.This bundle has structure group SO( m ). A spin structure s on E restrictsto an element of H ( O ; Z ) and determines a homomorphism π ( O ) → Z using (96). This can be used to construct a two-fold covering η : F → O andhence a principal bundle π ◦ η : F → M with structure group Spin( m ), whichis a lifting of the principal SO( m ) bundle O [17]. Note that the covering η contains the information about the spin structure [19].A bundle of spinors is a Hermitian vector bundle (a complex vector bundlewith a Hermitian inner product on each fibre) W associated to F . This isconstructed from the vector space C p with the standard Hermitian innerproduct, on which the m -dimensional Clifford algebra acts irreducibly. Thisis specified by { γ i , i = 1 , . . . , m } , a fixed set of anti-Hermitian p × p matrices,called gamma matrices, which satisfy γ i γ j + γ j γ i = − δ ij . (101)For even m , p = 2 m/ , while for odd m , p = 2 ( m − / .The gamma matrices determine a unitary action of Spin( m ) on C p whichis used to construct the spin bundle according to the associated bundle con-struction. The action of a spin transformation Z defines the correspondingrotation R = Φ( Z ) ∈ SO( m ) according to (cid:88) j ( R − ) ij γ j = Zγ i Z − (102)with ( R − ) ij the matrix components of R − .The Clifford module C p has some further properties. It has a real struc-ture, which is an antiunitary map j : C p → C p commuting with the action ofSpin( m ), with j = (cid:15) = ± m , it also has a chirality operator γ : C p → C p which is linear,unitary, satisfies γ = 1, and commutes with the action of Spin( m ). It relatesto the real structure by jγj − γ = (cid:15) (cid:48)(cid:48) = ±
1. The signs are determined by the27imension of the spin group m , mod 8. For even dimensions they are givenby the following table; for odd dimensions see [4]. m (cid:15) − − (cid:15) (cid:48)(cid:48) − − C p is the map ρ : ( R m ) ∗ → End( C p ) ρ ( ν ) = (cid:88) i ν i γ i (104)with ν i ∈ R the components of ν .These structures on C p carry over to similar properties of the spin bundle:there is a real structure j and chirality operator γ on each fibre W x . A Cliffordmultiplication is the action of cotangent vectors on W x , ρ x : T M ∗ x → End( W x ) , (105)satisfying at each point x the relations ρ x ( ν ) ρ x ( ν (cid:48) ) + ρ x ( ν (cid:48) ) ρ x ( ν ) = − g ( ν, ν (cid:48) ) (106) jρ x ( ν ) = ρ x ( ν ) j (107)for all ν and ν (cid:48) .A point f ∈ F x determines a linear isomorphism f : C p → W x called aspin frame. Explicitly, this is obtained by mapping ζ ∈ C p to the equivalenceclass [( f, ζ )] that defines a point of W x . This linear map determines the point f uniquely and so F is often called the bundle of spin frames.Denote the corresponding orthonormal frame e = η ( f ). The Cliffordmultiplication on the spinor bundle at point x is defined by ρ x ( ν ) = f ρ ( e ∗ ν ) f − . (108) Lemma 4.
The Clifford multiplication is independent of the choice of frame.
Proof.
The intertwining property (102) can be written ρ ( ν ) = Zρ ( R ∗ ν ) Z − . (109)If two spin frames differ by f (cid:48) = f Z , and hence e (cid:48) = eR , then this implies f ρ ( e ∗ ν ) f − = f (cid:48) ρ ( e (cid:48)∗ ν ) f (cid:48)− . (110)and so ρ x is independent of the choice of spin frame.28enote the smooth sections of a bundle by Σ and define the multiplicationmap µ : Σ( T ∗ M ⊗ W ) → Σ( W ) by( µ ( ν ⊗ w )) x = ρ x ( ν x ) w x . (111)A connection on the principal bundle O determines a unique connectionon F and hence a covariant derivative on sections of W . It will be assumedthat O has a connection. For the torus this will be the Levi-Civita connection,which is uniquely determined by the metric.If ∇ : Σ( W ) → Σ( T ∗ M ⊗ W ) is a covariant derivative on W , then theDirac operator on Σ( W ) is defined by Dψ = µ ∇ ψ. (112)This formalism can be extended to the case where the tangent bundle isaugmented with extra trivial directions R k and a spinor bundle W that ac-commodates the Clifford multiplication for the extra directions. The Cliffordmultiplication on this bundle isˇ ρ x : ( T M x ⊕ R k ) ∗ → End( W x ) , (113)and it is assumed that the extra directions have the standard metric andorientation, and are orthogonal to the tangent space. Then if P x : T M x ⊕ R k → T M x is the orthogonal projection, putting ρ x = ˇ ρ x P ∗ x (114)defines a Dirac operator again, using (112). The connection is again uniqueif one assumes that constant vectors in the extra directions are covariantlyconstant. Essentially, the extra directions act on the (possibly larger) spinorsbut play no role in the Dirac operator.Formula (108) generalises to this case, givingˇ ρ x ( ω ) = f ρ ( e ∗ ω ) f − (115)for the Clifford multiplication of ω ∈ ( T M x ⊕ R k ) ∗ , using a frame e for theSO( m + k ) bundle at point x , a frame f for the Spin( m + k ) bundle, and m + k gamma matrices. The Dirac operator can be written using a trivialisation of the orthonormalframe bundle O , i.e., a choice of an orthonormal frame e x for each point x of(a subset of) M . This is called a frame field.29dding trivial directions to the tangent bundle gives a greater flexibilityin the formulas that is important in this paper. This is because the framesdo not have to respect the direct sum decomposition of T M x ⊕ R k .Let ξ i be the standard basis vectors of R m + k . Then the orthogonal frame e determines m + k vector fields on M , denoted v i , by projecting onto thepart tangent to M . The definition is( v i ) x = P x e x ( ξ i ) . (116)The inverse metric tensor is given by the expression m + k (cid:88) i =1 v i ⊗ v i ∈ T M ⊗ T M . (117)The Dirac operator can be expressed in terms of the frame fields. Lemma 5.
Let
U ⊂ M be an open subset on which there is a frame field e : U → O covered by a spin frame field f : U → F . Then the Dirac operatoron sections of the spinor bundle restricted to U is Dψ = m + k (cid:88) i =1 f γ i f − ∇ v i ψ. (118) Proof.
The conclusion follows immediately from the formula µ = (cid:88) i v i ⊗ f γ i f − . (119)To prove this formula, suppose that ν ∈ Σ( T ∗ M ) and w ∈ Σ( W ). Then (cid:32)(cid:88) i v i ⊗ f γ i f − (cid:33) ( ν ⊗ w ) = (cid:88) i ν ( v i ) f γ i f − w. (120)At the point x this spinor is (cid:88) i ν x ( P x e x ξ i ) f x γ i f − x w x = (cid:88) i ( e ∗ x P ∗ x ν x )( ξ i ) f x γ i f − x w x = f x ρ ( e ∗ x P ∗ x ν x ) f − x w x = ˇ ρ ( P ∗ x ν x ) w x = ρ ( ν x ) w x = µ ( ν ⊗ w ) x , (121)using (116), (104), (115), (114), and (111).30 lobal frame fields From here on it will be assumed that the frame field e is global, i.e., exists smoothly over the whole of the manifold M . This isappropriate for the torus, and looks to be a useful geometric starting pointfor the construction of fuzzy spaces more generally.Suppose M is a Riemannian manifold, with its tangent space possiblyaugmented by trivial directions, as above. An orthonormal frame field e for T M ⊕ R k determines a particular spin structure s c called the canonical spinstructure. This is the spin structure such that e • ( s c ) = 0. The principalbundle of spin frames is the product bundle F = M ×
Spin( m + k ) and thetwo spin frames projecting to e are defined to be f = ±
1, with 1 standingfor the identity element of Spin( m + k ). The associated spinor bundle is asfollows. Definition 9.
Let e be an orthonormal frame field for T M ⊕ R k . Then thetrivial spinor bundle for the frame field e is defined to be the product vectorbundle W = M × C p . The Clifford multiplication on this bundle is definedaccording to (115) with f = 1.Using the same frame field, one can describe all of the other spin struc-tures on M . Let s be another spin structure, with the difference between s and s c parameterised by t ∈ H ( M ; Z ) as in (95). According to (96), theelement t determines a character˜ t : π ( M ) → {− , } ⊂ U(1) , (122)using the multiplicative action of Z on C . This can be used to construct theline bundle L ˜ t and hence a new spinor bundle W t = L ˜ t ⊗ W (123)corresponding to spin structure s . This gives a bundle of spinors W t that isperiodic or anti-periodic along a loop according to the value of ˜ t . Note thatthere may not be a spin frame field defined over the whole of M .In all these cases the spin frame on the universal covering space can betaken to be f = 1 and so the formula for the Dirac operator reduces to Dψ = m + k (cid:88) i =1 γ i ∇ v i ψ. (124) Example 11.
The circle R / π Z has a framing v = ∂∂θ . The one-dimensionalClifford algebra has two irreducible representations in C given by γ = i or − i . The bundle of spinors can be formed from R × C by one of the31wo quotients ψ ( θ + 2 π ) = ± ψ ( θ ), called periodic or anti-periodic boundaryconditions. The identification maps ± ∈ Spin(1) parameterise the two spinstructures; +1 is called the Lie group spin structure and − D = γ ∂∂θ . Example 12.
Augmenting the circle with one extra dimension leads to thefollowing Dirac operator depending on a fixed integer n . The orthonormalframe R → T S θ ⊕ R ∼ = R is e θ = (cid:18) cos nθ sin nθ − sin nθ cos nθ (cid:19) , (125)and hence the two vector fields v = cos nθ ∂∂θ , v = sin nθ ∂∂θ . (126)Using the gamma matrices γ = (cid:18) ii (cid:19) , γ = (cid:18) − (cid:19) , (127)the Dirac operator is D = γ ∇ v + γ ∇ v = (cid:18) ie − inθ ie inθ (cid:19) ∇ ∂∂θ , (128)and the chirality operator is γ = iγ γ = (cid:18) − (cid:19) . (129)As in Example 11, the bundle of spinors can be constructed from R × C with either periodic or anti-periodic boundary conditions.It is worth noting that the n = 0 cases of Example 12 are the same asthe direct sum of the two irreducible cases of Example 11 with the sameboundary conditions, after a change of basis in C . A gauge transformation is a function R : M →
SO( m + k ) that determinesa new frame field e (cid:48) = eR . The gauge transformation determines the homo-morphism R • : π ( M ) → π ( SO ( m + k )) . (130)32f m + k ≥ π ( SO ( m + k ) ∼ = Z , and if m + k <
3, then R • can becomposed with the stabilisation π ( SO ( m + k )) → π ( SO (3)) ∼ = Z givenby the inclusion SO ( m + k ) (cid:44) → SO (3). In either case one ends up with ahomomorphism R (cid:48)• : π ( M ) → Z . (131)If R (cid:48)• is the trivial homomorphism, R lifts to a function Z : M →
Spin( m + k ).For a general R , there is a lifting of it to a map on the universal coveringspace Z : (cid:99) M →
Spin( m + k ).Let W be the trivial spinor bundle for frame e using Definition 9. Define W (cid:48) to be the trivial spinor bundle for frame e (cid:48) and then define W (cid:48) t = L ˜ t ⊗W (cid:48) with ˜ t = ( − R (cid:48)• . Let µ be the Clifford multiplication of W and µ (cid:48) theClifford multiplication of W (cid:48) t . Lemma 6.
The map Σ( W (cid:48) t ) → Σ( W ) defined by ψ (cid:55)→ Zψ intertwines theClifford multiplications, i.e., µ (cid:48) = Z − µ (1 ⊗ Z ) . (132) Proof.
The computation is done on the covering space, where both bundlesare trivial and the spin frames are f = 1 and f (cid:48) = 1. In the followingequations, the point x is omitted to simplify the notation. µ (cid:48) ( ν ⊗ w ) = ρ (cid:48) ( ν ) w = ˇ ρ (cid:48) ( P ∗ ν ) w = ρ ( e (cid:48)∗ P ∗ ν ) w = ρ ( R ∗ e ∗ P ∗ ν ) w = Z − ρ ( e ∗ P ∗ ν ) Zw = Z − ˇ ρ ( P ∗ ν ) Zw = Z − ρ ( ν ) Zw = Z − µ ( ν ⊗ Zw ) . (133)Note that the third equality uses (115) and the fifth equality uses (109).The two spinor bundles have the same spin structure. The line bun-dle in the construction of W (cid:48) t cancels the change in homotopy class of theorthonormal frame field. Example 13.
In Example 12, a gauge transformation is defined by R ( θ ) = (cid:18) cos jθ sin jθ − sin jθ cos jθ (cid:19) (134)for a fixed integer j . The two lifts to Spin(2) ∼ = U(1) are ± e ijθ/ , representedin the Clifford module C by Z ( θ ) = ± (cid:18) e ijθ/ e − ijθ/ . (cid:19) (135)33f l denotes the circle, then R (cid:48)• ( l ) = j (mod 2). If j is odd, Z maps periodicspinors into anti-periodic spinors, and vice-versa.Applying the gauge transformation to the case n of Example 12 takes itto the case n + j , and in particular, it relates the case n = 0 to the case n = j . This shows that the spin structure is the Lie group structure if n iseven and the boundary conditions are periodic, or n is odd and the boundaryconditions anti-periodic. Otherwise it is the bounding spin structure.In general, the Dirac operator is preserved under a gauge transformation.More precisely, this is the following result. Lemma 7. If D is the Dirac operator on Σ( W ) and D (cid:48) the Dirac operatoron Σ( W (cid:48) t ), then D (cid:48) = Z − DZ . Proof.
A connection on the orthonormal frame bundle O determines a con-nection on any spin frame bundle uniquely, and hence a covariant derivativeon the associated bundle of spinors. The two spin frame bundles used toconstruct W and W (cid:48) t are isomorphic and therefore so are the associatedspinor bundles, with the isomorphism given in Lemma 6. Hence the covari-ant derivative on a section of W (cid:48) t in the direction of vector ξ is given by ∇ (cid:48) ξ ψ (cid:48) = Z − ∇ ξ Zψ (cid:48) , or more abstractly ∇ (cid:48) = (1 ⊗ Z − ) ∇ Z. (136)Then, using Lemma 6, D (cid:48) = µ (cid:48) ∇ (cid:48) = Z − µ (1 ⊗ Z ) (1 ⊗ Z − ) ∇ Z = Z − DZ. (137)
The description of the Dirac operator on a manifold is specialised to the caseof a torus. The spectrum of the Dirac operator is calculated for the generalflat torus using a parallel frame.Let A ∈ GL(2 , R ). Then a general parallel frame field on the torus isdetermined by the constant matrix A − = 1 ad − bc (cid:18) d − b − c a (cid:19) . (138)The spin structure is determined by σ ∈ Z × Z ∼ = H ( T ; Z ), as in Example10. The spinor bundle is W σ , as constructed in Section 5.3. This means that34 determines whether the spinor fields are periodic or anti-periodic alongeach of the two axes of the torus.The two cases of spinors of interest here are the irreducible case p = 2and the case p = 4. The Dirac operator is D = 1 ad − bc (cid:0) γ ( d∂ θ − c∂ φ ) + γ ( − b∂ θ + a∂ φ ) (cid:1) , (139)using two gamma matrices (labelled with 2 and 4 for later convenience). Thesquare of D is the Laplacian (78). The spectrum is calculated by substitutingthe plane waves ψ ( θ, φ ) = e i ( kθ + lφ ) ψ (140)into the Dirac equation and solving for the spinor ψ ∈ C p . The possiblevalues of k and l are either integer or half-integer and are determined by thespin structure, 2( k, l ) = σ (mod 2) . (141)The eigenvalue is given by the length of the vector ( k, l ) using the appropriatemetric [5], λ k,l, ± = ± ad − bc (cid:112) ( dk − cl ) + ( al − bk ) , (142)for ( k, l ) (cid:54) = (0 , p = 2, this eigenvalue has multiplicity onefor given values of k , l and ± . For the spin structure (0 , λ , = 0 with multiplicity two.The case p = 4 is the one of interest below. This occurs by introducingtwo additional gamma matrices γ and γ , as used for dimension m = 4. Thiscan be thought of as arising from the tensor product of the two-dimensionalspinors with another trivial two dimensional spinor module, C ∼ = C ⊗ C .For this case p = 4, all the above multiplicities are doubled.The spinors have a real structure j : C p → C p that commutes with thegamma matrices and satisfies the conditions (103). Extending j to an antilin-ear map J on spinor fields, it commutes with the Dirac operator, J D = DJ .If ψ is an eigenvector, then so is J ψ , which has the same eigenvalue. Thiscan be seen from (142), since λ k,l, ± = λ − k, − l, ± .The torus admits a U (1) × U (1) action by isometries, translating alongeach axis. This action is covered by an action of Spin(2) × Spin(2) on thespinor bundle. The elements of this group are parameterised by (Θ , Φ), eachvariable having period 4 π . The action on the spinor fields isΠ (Θ , Φ) ψ ( θ, φ ) = ψ ( θ + Θ , φ + Φ) . (143)This commutes with the Dirac operator and the plane waves in (140) areeigenvectors of P (Θ , Φ) with eigenvalue e i ( k Θ+ l Φ) .35 .6 Square torus The rotating frame field that generalises to the fuzzy case (in Section 6) isintroduced, giving a formula for the Dirac operator using this frame field.This is done first for the unit square torus, which is the case a = d = 1, b = c = 0. The rotating frame field uses the tangent space augmented with R . Thus there are four gamma matrices acting on spinors in C , and thereal structure j obeys the relations for dimension four in (103).The Dirac operator with the parallel frame specialises to D = γ ∂ φ + γ ∂ θ . (144)The parallel frame e : R → T ( T ) ⊕ R is defined by e ( ξ ) = ∂ φ , e ( ξ ) = ∂ θ ,and e ( ξ ) = (1 , e ( ξ ) = (0 ,
1) are basis vectors in the additional subspace R . Applying the SO(4) gauge transformation R ( θ, φ ) = cos θ sin θ − sin θ cos θ φ sin φ − sin φ cos φ (145)gives a new frame e (cid:48) = eR . The vector fields on T for the frame e (cid:48) are then w = − (sin θ ) ∂ φ w = (cos θ ) ∂ φ w = − (sin φ ) ∂ θ w = (cos φ ) ∂ θ (146)A lift of R to Spin(4) is given by Z = exp (cid:18) − (cid:0) θγ γ + φγ γ (cid:1)(cid:19) . (147)Due to the factor of 1 / Z is anti-periodic in both θ and φ .Therefore it determines a map of W σ to the bundle W (cid:48) σ +(1 , , following thediscussion in Section 5.4.Using Lemma 7, the Dirac operator on T with the rotating frame is D (cid:48) = Z − D Z = (cid:0) − (sin θ ) γ + (cos θ ) γ (cid:1)(cid:0) ∂ φ − γ γ (cid:1) + (cid:0) − (sin φ ) γ + (cos φ ) γ (cid:1)(cid:0) ∂ θ − γ γ (cid:1) . (148)Explicit formulas are given by choosing the gamma matrices γ = i
00 0 0 ii i γ = − − = i − i − i i γ = −
10 0 − . (149)The operator D (cid:48) has eigenvectors ψ k (cid:48) ,l (cid:48) , ± = e i ( k (cid:48) +1) θ + i ( l (cid:48) +1) φ e ik (cid:48) θ + il (cid:48) φ ± i l (cid:48) + k (cid:48) +1 √ ( k (cid:48) +1 / +( l (cid:48) +1 / e ik (cid:48) θ + i ( l (cid:48) +1) φ ± i ( k (cid:48) − l (cid:48) ) √ ( k (cid:48) +1 / +( l (cid:48) +1 / e i ( k (cid:48) +1) θ + il (cid:48) φ (150)and ψ k (cid:48) ,l (cid:48) , ± = ± i l (cid:48) + k (cid:48) +1 √ ( k (cid:48) +1 / +( l (cid:48) +1 / e i ( k (cid:48) +1) θ + i ( l (cid:48) +1) φ ± i ( k (cid:48) − l (cid:48) ) √ ( k (cid:48) +1 / +( l (cid:48) +1 / e ik (cid:48) θ + il (cid:48) φ e ik (cid:48) θ + i ( l (cid:48) +1) φ e i ( k (cid:48) +1) θ + il (cid:48) φ , (151)with (2 k (cid:48) + 1 , l (cid:48) + 1) = σ (mod 2). The corresponding eigenvalues are givenby ± (cid:112) ( k (cid:48) + 1 / + ( l (cid:48) + 1 / , (152)each with multiplicity 2. This agrees with (142) by setting k (cid:48) + 1 / k and l (cid:48) + 1 / l . Calculating the square of the rotating Dirac operator gives D (cid:48) = − ∂ θ − ∂ φ + γ γ ∂ θ + γ γ ∂ φ + 12 . (153)The covariant derivative on spinors determines a Laplace operator on spinors[22, Theorem 8.8]. In general, the Lichnerowicz-Schr¨odinger equation relatesthe square of the Dirac operator to this Laplacian and the curvature scalar.The curvature scalar is zero for a flat torus, so in this case the square of theDirac operator is equal to the Laplace operator on spinors. The case of the Dirac operator on a torus with a general integral metric androtating frame is now discussed. Consider the transformation A : ( θ, φ ) (cid:55)→ ( aθ + bφ, cθ + dφ ). This transformation is used to pull back functions on thesquare torus, giving C (cid:55)→ U = e i ( aθ + bφ ) , S (cid:55)→ V = e i ( cθ + dφ ) . (154)37ector fields are also related by pull-back, so for vector fields of the rotatingframe (146), v (cid:48) i = A • w i = A − • w i . Explicitly, these are v (cid:48) = 1 ad − bc ( b sin( aθ + bφ ) ∂ θ − a sin( aθ + bφ ) ∂ φ ) v (cid:48) = 1 ad − bc ( − b cos( aθ + bφ ) ∂ θ + a cos( aθ + bφ ) ∂ φ ) v (cid:48) = 1 ad − bc ( − d sin( cθ + dφ ) ∂ θ + c sin( cθ + dφ ) ∂ φ ) v (cid:48) = 1 ad − bc ( d cos( cθ + dφ ) ∂ θ − c cos( cθ + dφ ) ∂ φ ) . (155)Therefore, the lift to the action induced on the spin bundle W σ is Z = exp (cid:18) − (cid:0) γ γ ( aθ + bφ ) + γ γ ( cθ + dφ ) (cid:1)(cid:19) , (156)so that ψ (cid:48) = Z − ψ . The Dirac operator transforms to D (cid:48) = Z − DZ = 1 ad − bc (cid:0) b (sin( aθ + bφ ) γ − cos( aθ + bφ ) γ ) − d (sin( cθ + dφ ) γ − cos( cθ + dφ ) γ ) (cid:1)(cid:0) ∂ θ −
12 ( aγ γ + cγ γ ) (cid:1) + 1 ad − bc (cid:0) − a (sin( aθ + bφ ) γ − cos( aθ + bφ ) γ ) + c (sin( cθ + dφ ) γ − cos( cθ + dφ ) γ ) (cid:1)(cid:0) ∂ φ −
12 ( bγ γ + dγ γ ) (cid:1) (157)acting on sections of W (cid:48) σ (cid:48) , with σ (cid:48) = σ + (cid:0) [ a ] + [ c ] , [ b ] + [ d ] (cid:1) . It is easy tocheck that one recovers D (cid:48) when a = d = 1 , b = c = 0.The transformation A acts on the spinor fields by regarding them as C -valued functions on R and using the pull-back of functions, denoted A • .(The spinor components are not transformed.) The Dirac operator D (cid:48) canbe characterised by this pull-back. It obeys D (cid:48) A • = A • D (cid:48) . (158)The spinor Laplacian for the general integral torus in the rotating frameis ( D (cid:48) ) = − ad − bc ) (cid:18) ( b + d ) ∂ ∂θ − ab + cd ) ∂ ∂θ∂φ + ( a + c ) ∂ ∂φ (cid:19) + 1 ad − bc ( γ γ ( d∂ φ − c∂ θ ) + γ γ ( a∂ φ − b∂ θ )) + 12 . (159)38he Spin(2) × Spin(2) action in the rotating frame becomesΠ (cid:48) (Θ , Φ) ψ (cid:48) ( θ, φ ) = W ψ (cid:48) ( θ + Θ , φ + Φ) , (160)with a non-trivial gauge transformation W = Z (Θ , Φ) = exp (cid:18) −
12 ( γ γ ( a Θ + b Φ) + γ γ ( c Θ + d Φ)) (cid:19) (161)on the spinors. The Dirac operator D (cid:48) then commutes with the action ofSpin(2) × Spin(2) on the spinor fields, i.e., D (cid:48) Π (cid:48) (Θ , Φ) = Π (cid:48) (Θ , Φ) D (cid:48) . (162) The Dirac operator on a fuzzy space is the fundamental structure that en-codes the geometry of the space. It is characterised by a set of algebraicaxioms for a structure called a finite real spectral triple. This section intro-duces a particular spectral triple for each fuzzy torus. This gives a Diracoperator for the non-commutative analogues of the tori with integral metricsand arbitrary spin structures, as presented in Section 5.7.Real spectral triples were originally defined in [1] and [3], the axioms un-dergoing some slight modifications since then. The definition includes com-pact spin manifolds in the commutative case. Specialising to the finite casesimplifies the definition because the analytic axioms are not required. Theprecise definition of the finite case is the one given in [4] and is summarisedbriefly here for the even-dimensional cases.
Definition 10.
A finite real spectral triple is • An integer d defined mod 8, called the KO-dimension. • A finite-dimensional Hilbert space H . • A ∗ -algebra A of operators with a faithful left action in H . • A Hermitian operator Γ :
H → H , called the chirality, that commuteswith the algebra action and obeys Γ = 1. • An antiunitary map J : H → H , called the real structure, such that[
J aJ − , b ] = 0 (163)for all a, b ∈ A . The real structure satisfies J = (cid:15) , J Γ J − Γ = (cid:15) (cid:48)(cid:48) , using(for even d ) the sign table in Section 5.2.39 A Hermitian operator D : H → H , called the Dirac operator, satisfying[[
D, a ] , J bJ − ] = 0 (164)for all a, b ∈ A , and (for even d ), D Γ + Γ D = 0 and DJ = J D .The axioms determine a right action of a ∈ A on ψ ∈ H by ψ a = J a ∗ J − ψ. (165)This commutes with the left action and so makes H a bimodule.The fuzzy analogue presented here is of a flat torus with a tangent spaceaugmented with two extra dimensions and a non-constant framing. This usesthe space of spinors C regarded as a Clifford module of type (0 , γ = γ γ γ γ and the real structure is j v v v v = v − v − v v , which satisfies jγ i = γ i j .The following definition is phrased in a general way but all of the examplesthat follow will have a very simple construction in terms of spaces of matrices. Definition 11.
Let
U, V, h be a fuzzy torus with real structure J : h → h .Also, let X, Y ∈ (cid:104)
U, V (cid:105) with XY = QY X and choose a fourth root Q / .The spectral triple for the fuzzy torus is as follows. • The KO-dimension is 4. • The Hilbert space is H = C ⊗ h . • The ∗ -algebra is A = (cid:104) U, V (cid:105) , acting in H on the left by U ( v ⊗ m ) = v ⊗ U m, V ( v ⊗ m ) = v ⊗ V m • The real structure is J ( v ⊗ m ) = jv ⊗ J m . • The chirality operator is Γ( v ⊗ m ) = γv ⊗ m .40 The Dirac operator on the fuzzy torus is D X,Y = − Q / − Q − / ) ( γ ⊗ [ X + X ∗ , · ] + γ ⊗ i [ X ∗ − X, · ] − γ ⊗ [ Y + Y ∗ , · ] − γ ⊗ i [ Y ∗ − Y, · ]) − Q / + Q − / ) ( γ γ γ ⊗ { X + X ∗ , · } − γ γ γ ⊗ i { X ∗ − X, · } + γ γ γ ⊗ { Y + Y ∗ , · } − γ γ γ ⊗ i { Y ∗ − Y, · } ) , using the right action determined by J .It is a straightforward calculation to check that this defines a real spectraltriple. In particular, the first order condition (164) follows from the fact that D is a sum of terms that commute with either the left action or the rightaction of U and V .The Dirac operator in Definition 11 may be written a little more system-atically, as in [4], as D X,Y = 1 Q / − Q − / (cid:88) i γ i ⊗ [ K i , · ] + 1 Q / + Q − / (cid:88) i 14 ( X + X ∗ ) , K = − K = − i X ∗ − X ) ,K = − K = 14 ( Y + Y ∗ ) , K = K = i Y ∗ − Y ) . (167)The most delicate aspects of this definition are the numerical coefficientsinvolving Q / . Some insight into these is obtained by calculating the squareof the Dirac operator. A lengthy calculation shows D X,Y = − Q / − Q − / ) (1 ⊗ [ X, [ X ∗ , · ] + 1 ⊗ [ Y, [ Y ∗ , · ])+ i Q / − Q − / ) ( γ γ ⊗ { Y, [ Y ∗ , · ] } − γ γ ⊗ { X, [ X ∗ , · ] } )+ 14( Q / + Q − / ) (1 ⊗ { X, { X ∗ , ·}} + 1 ⊗ { Y, { Y ∗ , ·}} ) . (168)The calculation of the square can be broken down in the following way.Setting E X = − Q / − Q − / ) ( γ ⊗ [ X + X ∗ , · ] + γ ⊗ i [ X ∗ − X, · ]) − Q / + Q − / ) ( γ γ γ ⊗ { X + X ∗ , · } − γ γ γ ⊗ i { X ∗ − X, · } )41nd E Y = − Q / − Q − / ) ( − γ ⊗ [ Y + Y ∗ , · ] − γ ⊗ i [ Y ∗ − Y, · ]) − Q / + Q − / ) ( γ γ γ ⊗ { Y + Y ∗ , · } − γ γ γ ⊗ i { Y ∗ − Y, · } ) . Then D X,Y = E X + E Y . The crucial property is E X E Y + E Y E X = 0 (169)so that D X,Y = E X + E Y . The property (169) does actually fix the numericalcoefficients uniquely, up to an overall constant. It is worth noting that theformulas for E X and E Y correspond if one interchanges γ ↔ γ , γ ↔ γ , X ↔ Y and Q / ↔ Q − / . To calculate explicit features of the fuzzy torus Dirac operator one mustconsider specific choices of X and Y . This corresponds to picking a specificmetric on a fuzzy torus. Mirroring Section 5, the square fuzzy torus is con-sidered first. This is the simplest case, allowing a discussion of the mainfeatures of the Dirac operator without introducing a lot of formalism. Moregeneral cases are discussed in Sections 6.2 and 6.3.Let h = M N ( C ), J = ∗ , X = U = C and Y = V = S , so that A = M N ( C ).The first thing to note is that the Dirac operator corresponds to thecommutative case (148) using the heuristics (75) for the commutators, adding { C, ·} (cid:32) e iθ , { C ∗ , ·} (cid:32) e − iθ { S, ·} (cid:32) e iφ , { S ∗ , ·} (cid:32) e − iφ (170)for the anticommutators, then setting q / to 1.Define the vector ψ = α kl e ( k +1 ,l +1) β kl e ( k,l ) γ kl e ( k,l +1) δ kl e ( k +1 ,l ) , (171)with α kl , β kl , γ kl , δ kl fixed complex numbers, each a function of integers k and l . This is an eigenvector of the Dirac operator, the eigenvalue equation being D C,S ψ = ([ k + 1 / q + [ l + 1 / q ) ψ. (172)42igure 3: A 3D plot of the positive eigenvalues (173) with N = 100.Each eigenvalue has multiplicity 4, as expected from the spinor doubling. Itfollows immediately that the eigenvalues of D C,S are given by λ k,l, ± = ± (cid:113) [ k + 1 / q + [ l + 1 / q (173)with each eigenvalue having multiplicity 2.The eigenspaces are periodic in the labels k , l , and so a unique labellingis given by − N < k, l ≤ N . (174)Plotting (173) for a fixed value of N shows the geometry of the fuzzy torus.As in Section 4, the value q = e πi/N is used for all of the following plots.Figure 3 shows the dispersion relation, the discrete values for integer k and l being interpolated by a smooth surface. This surface would be periodic if k and l were continued to Z × Z , and looks like a dispersion relation for amassless electron in a lattice in solid state physics.Taking the limit q / → k and l fixed gives[ k + 1 / q + [ l + 1 / q → ( k + 1 / + ( l + 1 / . (175)These are the eigenvalues for the commutative spinor Laplacian on the flatsquare torus with spin structure σ = (1 , N = 100 and k, l > σ = (1 , 1) is shown in Figure 4, where a portion of each surface is shown side-by-side. Notice that for small values of k and l , the spectra of the fuzzy torusand commutative torus are practically identical. Physically, this means thatone cannot detect the non-commutative behaviour of the fuzzy torus at lowenergies.It is also useful to represent the spectrum through a contour plot. Thiswill allow an easy comparison of the geometries of different tori. The contourplot for the square fuzzy torus is shown in Figure 5. The plot has some re-markably symmetrical features that encode information about the geometryand the multiplicities. The lines on which the large multiplicity eigenvalueslie are defined by ( l + 1 / ± ( k + 1 / 2) = ± (cid:48) ( N/ jN ) with j ∈ Z . Thelarge multiplicities appear due to a relation similar to (86) being satisfied.A comparison of the multiplicities of the commutative torus and the fuzzytorus is shown in the histogram in Figure 6. The commutative torus mul-tiplicities follow an approximate V shape until the artificial cut-off (174) isreached. One sees that for small eigenvalues, the fuzzy torus and commuta-tive torus have approximately the same multiplicities but diverge from eachother at higher energies. 44 - 20 0 20 40 - - k l Figure 5: Contour plot for the positive eigenvalues (173) with N = 100. Fuzzy Torus Commutative Torus - - - 20 0 20 40 60 λ Figure 6: Histogram of multiplicity against the eigenvalues (173) and (142),with N = 100 and bin width 2. 45 .2 Fuzzy torus with integral metric An integral metric is determined by the choice of X and Y . Similarly toSection 4.1, let h = M N ( C ), J = ∗ , U = C and V = S , so that A = M N ( C ).Now set X = E ( a,b ) , Y = E ( c,d ) , so that XY = QY X with Q / = q ( ad − bc ) / .Note that different choices of U and V are possible, as will be discussed inSection 6.3. With the choice given here the fuzzy torus is irreducible.The action of the group Z N × Z N on the fuzzy torus algebra, as given inDefinition 4, is the non-commutative analogue of the group action U (1) × U (1)on the commutative torus. This action can be extended to the spectral triple.It is shown here that the fuzzy torus Dirac operator is equivariant under thisaction and that this is analogous to the explicit formulas for the equivarianceof the commutative integral Dirac operator shown in Section 5.7.Let q / = e iπK/N . For j, n ∈ Z , defineΘ = 2 πKjN , Φ = 2 πKnN (176)and W = exp (cid:18) − 12 ( γ γ ( a Θ + b Φ) + γ γ ( c Θ + d Φ)) (cid:19) . (177)The action on the Hilbert space of the spectral triple isΠ ( j,n ) ( v ⊗ m ) = W v ⊗ P ( j,n ) m P − j,n ) , (178)with P ( j,n ) given in Definition 4. This is the non-commutative analogue of(160). The fuzzy torus Dirac operator is equivariant with respect to thisaction, i.e., D X,Y Π ( j,n ) = Π ( j,n ) D X,Y . (179)Since the eigenvalues of γ γ and γ γ are ± i , the action is periodic. If N is odd Π ( N, = 1 , Π (0 ,N ) = 1 , (180)whereas if N is even,Π ( N, = ( − a + c , Π (0 ,N ) = ( − b + d . (181)Thus Π ( j,n ) is a projective representation of Z N × Z N .Define the canonical spin structure for this geometry σ c = ([ a ] + [ c ] , [ b ] + [ d ] ) = [ A T ] (1 , , (182)46ccording to (100). Then parameters k and l are defined by 2( k, l ) = σ c (mod 2), as in (141). These parameters determine vectors ψ = α e ( k +( a + c ) / , l +( b + d ) / β e ( k − ( a + c ) / , l − ( b + d ) / γ e ( k +( c − a ) / , l +( d − b ) / δ e ( k +( a − c ) / , l +( b − d ) / , (183)with α, β, γ, δ arbitrary complex numbers. A short calculation shows theseare eigenvectors for the action of the translations, i.e.,Π ( j,n ) ψ = e i (Θ k +Φ l ) ψ. (184)The ψ are also eigenvectors of D X,Y , the eigenvalue equation being D X,Y ψ = (cid:18) [ al − bk ] q + [ dk − cl ] q [ ad − bc ] q (cid:19) ψ = (cid:32)(cid:20) al − bkad − bc (cid:21) Q + (cid:20) dk − clad − bc (cid:21) Q (cid:33) ψ. (185)One can find the eigenvectors of D X,Y itself by solving a 4 × α, β, γ, δ . The corresponding eigenvalues are the twosquare roots of the eigenvalues in (185), each with multiplicity two. Theseformulas for the eigenvectors and eigenvalues are direct analogues of the com-mutative case, for example as shown explicitly for the square torus in Section5.6.Now suppose that k and l are both integers, i.e., σ c = 0. Then theeigenvalues are exactly the same as for the scalar Laplacian in (76). Thereare in fact two natural unitary transformations relating 1 ⊗ ∆ U,V and D U,V .Define algebra elements T = E ( a + c , b + d ) and H = E ( c − a , d − b ) . Using thedefining (left) action of these operators in H , the first unitary transformationis U L = 14 ( i ( γ γ − γ γ )( H ∗ − H ) − i ( γ γ + γ γ )( T ∗ − T )+ (1 + γ )( H + H ∗ ) + (1 − γ )( T + T ∗ )) , (186)while the second one is U R = J U L J − . (187)Then 1 ⊗ ∆ X,Y = U L D X,Y U ∗ L = U R D X,Y U ∗ R . (188)One may think of both U L and U R as non-commutative analogues of the spintransformation Z introduced in (156). In fact, the formula (186) becomesexactly (156) if T and H are replaced by the corresponding exponentialfunctions of the variables θ and φ . 47 - 20 0 20 40 - - k l Figure 7: Contour plot for the spectrum (189) with N = 100. Example 14. Let X = C , Y = S so that XY = QY X and Q = q . Theeigenvalues of the Dirac operator are λ k,l, ± = ± (cid:115)(cid:20) k + 12 (cid:21) Q + (cid:20) l + 12 (cid:21) Q . (189)The eigenvalues are presented in a contour plot in Figure 7. It can be seenthat the torus has now been stretched equally in both directions by a factorof two. The large eigenvalue multiplicities analogous to (86) lie on the linesdefined by ( l + 1) ± ( k + 1) = ± (cid:48) ( N/ jN ) for j ∈ Z . Example 15. Let X = q − / CS, Y = S and Q = q . The eigenvalues ofthe Dirac operator are λ k,l, ± = (cid:115)(cid:20) l − k + 12 (cid:21) Q + (cid:20) k + 12 (cid:21) Q . (190)Again, a contour plot of these eigenvalues, in Figure 8, gives a visual repre-sentation of the geometry. This plot shows clearly the non-square geometryof this torus. Section 6.2 showed how to construct a non-commutative spectral triple foreach integral metric on the torus and a value of N . The commutative ana-logue of this has a particular spin structure, the canonical spin structure. In48 - 20 0 20 40 - - k l Figure 8: Contour plot for the eigenvalues (190) and N = 100.this section, it is shown how to construct a spectral triple for a fuzzy toruswith any integral metric and any spin structure. This is done by construct-ing a non-commutative generalisation of the appropriate covering space of atorus.The definitions could be phrased more generally but since it is not yetclear whether there are other interesting examples, this is left as an openproblem. There is a very interesting analogy with symplectic reduction,which is outlined at the end of this section, and may help with finding theappropriate non-commutative context.The commutative construction that is to be generalised is as follows.There is an analogue of the Hurewicz homomorphism for Z coefficients.This is the homomorphism h : π ( M ) → H ( M ; Z ) that takes a loop to thecorresponding Z -cycle. It is obtained from the usual Hurewicz homomor-phism h : π ( M ) → H ( M ; Z ) by tensoring with 1 ∈ Z . A Riemannianmanifold M has a regular covering space (cid:99) M associated to h ; this is a prin-cipal bundle with group H ( M ; Z ). Now suppose M has spin structure s .The spinor bundle W on M pulls back to a spinor bundle (cid:99) W on (cid:99) M , witha pull-back Clifford multiplication, and hence a uniquely-determined Diracoperator (cid:98) D . Sections of (cid:99) W that are equivariant with respect to a character χ : H ( M ; Z ) → U(1), i.e., gψ = χ ( g ) ψ (191)can be considered sections of a spin bundle on M with another spin structure s (cid:48) . This spin structure is determined by s (cid:48) = s + π • t , with t ∈ H ( M ; Z ) the49ohomology class that corresponds to χ , i.e., χ = ( − t . Then (cid:98) D restrictedto these sections gives the Dirac operator on M with the spin structure s (cid:48) . Non-commutative coverings The basic structure is an inclusion of ∗ -algebras A ⊂ B . In the commutative case, this is the inclusion of functionsthat is dual to the projection map of the covering.For the non-commutative example, both A and B are fuzzy tori. It sufficesto consider the case where the ‘total space’ is the fuzzy torus (cid:0) C, S, h = M N ( C ) (cid:1) with clock and shift operators satisfying the relation CS = q SC ,and with real structure J = ∗ . Then B = (cid:104) C, S (cid:105) = M N ( C ).Now suppose U, V ∈ B satisfying U V = qV U , with N (cid:48) the order of q .Then A = (cid:104) U, V (cid:105) is the algebra for the second fuzzy torus ( U, V, h ) formingthe ‘base space’. The elements U N (cid:48) and V N (cid:48) are central in A = (cid:104) U, V (cid:105) andso generate a finite abelian group G .This group acts in h by the adjoint action ψ (cid:55)→ gψg − , and as automor-phisms of B by b (cid:55)→ gbg − . Since elements of the subalgebra A are invariantunder this action, it is a non-commutative analogue of the deck transforma-tions of a covering. Note that G acts non-trivially on the algebra B : forexample, U N (cid:48) SU − N (cid:48) = − S .The Hilbert space h splits into subspaces according to the unitary char-acters of G , i.e., h χ is the subspace of vectors ψ satisfying gψg − = χ ( g ) ψ, (192)a non-commutative version of (92). Then h = ⊕ χ h χ is the non-commutativeversion of (93). Acting with J shows g ( J ψ ) g − = J ( gψg − ) = J χ ( g ) ψ = χ ( g ) J ψ, (193)so that J ψ lies in h χ , with χ the complex conjugate character.In particular, J ψ lies in h χ if χ = χ , which is true if all elements of G have order two. In this case, each ( U, V, h χ ) is a fuzzy torus. Non-commutative spin structures Specialising to the case of that isanalogous to spin structures, it is necessary to have U N (cid:48) (cid:54) = 1, V N (cid:48) (cid:54) = 1 andthe relations U N (cid:48) = V N (cid:48) = 1, so that G ∼ = Z × Z . This can be achievedby setting N = 4 N (cid:48) and U = C , V = S (194)to define A ⊂ B . The parameters are related by q = q .According to Example 9, A = A ⊕ A ⊕ A ⊕ A and, moreover,each summand is a ∗ -algebra. There is also a surjective homomorphism A → A ∼ = M N (cid:48) ( C ) = (cid:104) C (cid:48) , S (cid:48) (cid:105) given by U (cid:55)→ C (cid:48) , V (cid:55)→ S (cid:48) .50pectral triples are constructed according to Definition 11 using the Diracoperator D X,Y , for X, Y ∈ A . This defines spectral triples for the fuzzy torus (cid:0) C, S, h (cid:1) , with algebra B , and also for the fuzzy torus ( U, V, h ), with algebra A . Since the characters are all real, χ = χ , the latter splits into fuzzy tori( U, V, h χ ), giving spectral triples with Hilbert space H χ = C ⊗ h χ .The four characters of G are labelled 00 , , , 11 and the splitting iswritten H = H ⊕ H ⊕ H ⊕ H , (195)in a similar way to Example 9, corresponding to the bundles with periodic oranti-periodic boundary conditions in the commutative analogue. Using thisanalogy, the spin structure of T associated with H χ should be defined as σ = σ c + t , with χ = ( − t . Example 16. The simplest example is for the square fuzzy torus determinedby D U,V . The eigenvectors of D U,V are given by (183) with a = d = 2, b = c =0. They are indexed by integers k and l and lie in H hj if ( k + 1 , l + 1) = ( h, j )mod 2.The spectrum of D U,V is readily calculated from (185). The eigenvalue of D U,V is (cid:20) l (cid:21) q + (cid:20) k (cid:21) q . (196)One can see that the eigenvalues are exactly the same as in (172) in the case h = j = 0, so that the spin structure is σ = (1 , h, j , the eigenvaluescorrespond to the commutative case (152) if σ = ( h + 1 , j + 1). Thus oneidentifies σ c = (1 , 1) as expected. Symplectic reduction As a final remark, the algebra homomorphisms A ← A (cid:44) → B form a discrete and non-commutative analogue of the mapsin a Marsden-Weinstein reduction of a symplectic manifold by a group action[23]. There, starting with a Lie group G and a Hamiltonian G -action on asymplectic manifold M with momentum map µ , one has the maps M // G (cid:44) →M /G ← M , with M // G = µ − (0) /G .In the non-commutative example, the analogue of G is the finite group G . The algebra B is the non-commutative analogue of (functions on) M , A is the analogue of the space of orbits M /G , while A is the analogue of thesymplectic quotient M // G .Denote the group algebra of G by C [ G ]. This is dual to the group of uni-tary characters G ∗ , i.e., C [ G ] ∼ = C ( G ∗ ). The inclusion C [ G ] (cid:44) → B is the ana-logue of the momentum map µ , the (discrete) momenta being the elements51f G ∗ . Indeed, the ‘constraint space’ A corresponds exactly to momentum1 ∈ G ∗ . References [1] A. Connes, “Noncommutative geometry and reality,” J. Math. Phys. (1995) 6194. doi:10.1063/1.531241[2] A. Connes, “Noncommutative geometry,” Academic Press, Inc., SanDiego, CA, 1994, xiv+661, 0-12-185860-X.[3] A. Connes, “Gravity coupled with matter and foundation of non-commutative geometry,” Commun. Math. Phys. (1996) 155doi:10.1007/BF02506388 [hep-th/9603053].[4] J. W. Barrett, “Matrix geometries and fuzzy spaces as finite spectraltriples,” J. Math. Phys. (2015) no.8, 082301 doi:10.1063/1.4927224[arXiv:1502.05383 [math-ph]].[5] C. B¨ar, “Dependence of Dirac Spectrum on the Spin Structure,”In: Global analysis and harmonic analysis (Marseille-Luminy, 1999),S´eminaires and Congr´es, , J.P. Bourguignon, T. Branson, O. Hijazi,(eds.), Soc. Math. France, Paris (2000) pp. 17-33. [arXiv:math/0007131][6] G. Landi, F. Lizzi and R. J. Szabo, “From large N matrices tothe noncommutative torus,” Commun. Math. Phys. (2001) 181doi:10.1007/s002200000356 [hep-th/9912130].[7] P. Schreivogl and H. Steinacker, “Generalized Fuzzy Torus and its Mod-ular Properties,” SIGMA (2013) 060 doi:10.3842/SIGMA.2013.060[arXiv:1305.7479 [hep-th]].[8] Y. Kimura, “Noncommutative gauge theories on fuzzy sphere andfuzzy torus from matrix model,” Prog. Theor. Phys. (2001) 445doi:10.1143/PTP.106.445 [hep-th/0103192].[9] R. Høegh-Krohn, T. Skjelbred, “Classification of C ∗ -algebras admittingergodic actions of the two-dimensional torus”, Journal f¨ur die Reineund Angewandte Mathematik. [Crelle’s Journal], , (1981), 1–8, DOI10.1515/crll.1981.328.1[10] M. Paschke and A. Sitarz, “On Spin Structures and Dirac Opera-tors on the Noncommutative Torus,” Lett. Math. Phys. (2006) 317doi:10.1007/s11005-006-0094-2 [math/0605191 [math.QA]].5211] J. J. Venselaar, “Classification of spin structures on the non-commutative n-torus,” J. Noncommut. Geom. (2013) no.3, 787doi:10.4171/JNCG/135 [arXiv:1003.5156 [math.OA]].[12] A. Carotenuto and L. Dabrowski, “Spin geometry of the ra-tional noncommutative torus,” J. Geom. Phys. (2019) 28doi:10.1016/j.geomphys.2019.05.008 [arXiv:1804.06803 [math.QA]].[13] J. W. Barrett “Non-commutative spectral triples for space-time.” In:Mathematisches Forschungsinstitut Oberwolfach Report No. 32/2018Non-commutative Geometry, Index Theory and Mathematical Physics,8–14 July 2018 eds. A. Connes, R. Nest, T. Schick and G. Yu. 43–45DOI: 10.4171/OWR/2018/32[14] J. Gaunt, “Aspects of the noncommutative torus”. PhD thesis, Univer-sity of Nottingham. (2019) http://eprints.nottingham.ac.uk/56288/[15] H. Weyl, “The theory of groups and quantum mechanics,” Translatedfrom the second (revised) German edition by H. P. Roberton; Reprintof the 1931 English translation, Dover Publications, Inc., New York,(1950)[16] G.W. Mackey, “Some remarks on symplectic automor-phisms,” Proc. Amer. Math. Soc. 16 (1965), 393-397 DOI:https://doi.org/10.1090/S0002-9939-1965-0177064-5[17] J. W. Barrett, “Holonomy and path structures in general relativ-ity and Yang-Mills theory,” Int. J. Theor. Phys.30